Question

If A (1,2), B (4, y), C (x,6) and D (3,5) are vertices of a parallelogram ABCD, find the values of $x$ and $y$. [4 MARKS]

Answer 1

Formula: 1 Mark
Steps: 1 Mark
Value of each $x$ and $y$: 1 Mark each

Consider a parallelogram ABCD .

Let O be the point of intersection of the diagonals AC and BD

We know that diagonals of a parallelogram bisect each other.

$\Rightarrow$ O is the midpoint of both the diagonals AC and BD

Now, coordinates of O as mid-point of BD are

$O(x,y)=\left(\frac{x_1+x_2}{2}\right),\left(\frac{y_1+y_2}{2}\right)$

$\Rightarrow O(a,b)=\frac{4+3}{2},\frac{y+5}{2}\dots \dots \left(1\right)$

Also, coordinates of O as mid-point of AC are

$O(a,b)=\frac{1+x}{2},\frac{2+6}{2}\dots \dots \left(2\right)$

From (1) and (2), we have

$\frac{4+3}{2}=\frac{1+x}{2}$

$\frac{7}{2}=\frac{1+x}{2}\Rightarrow x=6$

And $\frac{y+5}{2}=\frac{2+6}{2}$

$\frac{y+5}{2}=\frac{8}{2}\Rightarrow y=3$